Enhanced Estimation of the Population Mean Using Two Auxiliary Variables under Probability Proportional to Size Sampling

<div>Family members of the suggested generalized estimators <span class="nowrap"><svg height="23.7426pt" id="M41" style="vertical-align:-5.3319pt" version="1.1" viewbox="-0.0498162 -18.4107 37.4029 23.7426" width="37.4029pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,1.187,-6.159)"><path d="M470 557C393 591 314 624 246 666H224C156 624 76 591 0 557L12 532C86 550 163 573 235 596C309 572 384 550 458 532L470 557Z"></path></g><rect height="0.65243" width="8.50777" x="0" y="-11.1958"></rect><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M640 650H413L406 622L438 617C475 611 483 605 463 571C422 503 346 400 296 333C264 403 215 529 199 575C189 604 191 614 228 618L262 622L271 650H30L23 622C90 615 97 609 119 553L212 321C222 295 223 285 218 259L193 127C176 39 167 33 82 28L76 0H350L356 28C266 36 257 40 275 127L303 266C308 290 314 302 328 322C396 418 454 493 498 546C551 610 565 616 631 622L640 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.508,-11.894)"><path d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 262C225 236 198 227 145 220C133 201 125 177 126 158C141 148 167 143 189 144C222 186 243 205 290 236C288 179 282 152 262 102C272 82 289 63 306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,14.068,-11.894)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,17.326,-11.894)"><path d="M465 45L455 77C444 72 433 69 421 69C404 69 389 81 389 128V293C389 349 380 392 337 425C310 445 280 454 239 454C182 443 111 409 73 383C51 366 38 353 38 335C38 309 74 285 101 285C121 285 136 293 143 319C151 350 160 374 171 388C180 400 191 406 205 406C241 406 265 371 265 308V278C242 258 165 234 106 212C54 192 30 168 30 116C30 54 81 -12 165 -12C185 -12 232 21 270 46C274 34 281 21 295 9C308 -3 329 -12 356 -12L465 45ZM265 79C249 67 238 63 223 63C200 63 163 84 163 139C163 174 179 192 210 208C231 219 247 229 265 239V79Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,21.582,-11.894)"><path d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,23.739,-11.894)"><path d="M196 417V703C139 687 58 673 7 667V636C63 632 68 623 68 566V24C137 -2 188 -12 244 -12C390 -12 504 97 504 243C504 355 418 454 297 454C277 454 261 447 233 434L196 417ZM196 382C211 387 221 389 238 389C301 389 365 339 365 212C365 87 316 27 262 28C230 29 207 59 201 75C198 83 196 94 196 108V382Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,28.616,-11.894)"><path d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.508,3.784)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.766,3.784)"><path d="M463 633C453 636 427 644 417 648C388 660 349 667 316 667C183 667 100 592 100 483C100 398 176 335 223 305L259 282C321 242 358 203 358 149C358 66 301 22 234 22C110 22 70 131 57 208L24 205L28 48C51 23 128 -17 215 -17C342 -17 446 60 446 174C446 262 394 310 317 361L284 383C229 420 183 455 183 515C183 569 222 628 298 628C384 628 418 563 428 481L461 487C459 527 458 593 463 633Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,16.116,3.784)"><path d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,18.272,3.784)"><path d="M622 479C622 589 545 650 400 650H146L140 618C229 609 230 602 215 527L135 132C117 43 112 41 27 32L18 0H305L313 32C213 40 205 44 222 132L251 284L324 262H365C484 269 622 333 622 479ZM528 477C528 356 420 305 344 305C300 305 276 314 259 321L303 554C311 597 313 600 331 606C345 611 361 614 379 614C463 614 528 580 528 477Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,23.434,3.784)"><path d="M622 479C622 589 545 650 400 650H146L140 618C229 609 230 602 215 527L135 132C117 43 112 41 27 32L18 0H305L313 32C213 40 205 44 222 132L251 284L324 262H365C484 269 622 333 622 479ZM528 477C528 356 420 305 344 305C300 305 276 314 259 321L303 554C311 597 313 600 331 606C345 611 361 614 379 614C463 614 528 580 528 477Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,29.097,3.784)"><path d="M463 633C453 636 427 644 417 648C388 660 349 667 316 667C183 667 100 592 100 483C100 398 176 335 223 305L259 282C321 242 358 203 358 149C358 66 301 22 234 22C110 22 70 131 57 208L24 205L28 48C51 23 128 -17 215 -17C342 -17 446 60 446 174C446 262 394 310 317 361L284 383C229 420 183 455 183 515C183 569 222 628 298 628C384 628 418 563 428 481L461 487C459 527 458 593 463 633Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,33.447,3.784)"><path d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"></path></g></svg>.</span></div>

Mathematical Problems in Engineering

tab1

Table 1

Table 1: Enhanced Estimation of the Population Mean Using Two Auxiliary Variables under Probability Proportional to Size Sampling 